Data is taken from these resources:
Analysis was proceeded using this statistical sources:
Main target is to get result table containing all “expected” relevant data/variables that can be connected with changes of demand in Kelheim region, thus mobility data is collected on daily basis, remaining data has to be daily based.
After obtaining such table it would be useful(in sense of building a model and understanding the data) to check correlations between individual independent variable to dependent mobility variable. So the next chapter has a focus on filtering out unrelevant data to pass it to the model.
##Modify and join data
# Weatherstack
weatherstack_kelheim_daily = weatherstack_kelheim %>%
group_by(date) %>%
count(description)
# Stringency
deu_stringency = json[grep("DEU.stringency_actual",names(json))]
date_stringency = sapply(strsplit(names(deu_stringency),split = ".",fixed = TRUE),"[[",2)
df_stringency = data.frame(date = date_stringency,stringency = deu_stringency)
df_stringency = df_stringency %>% mutate(stringency = as.numeric(stringency), date = as.Date(date))
# Ingolstadt
type_of_weather = unique(weatherstack_kelheim$description)
map_vector <- c("Clear","Sunny","Cloudy","Light","Light","Light","Light","Light","Light","Light","Light","Medium","Cloudy","Light","Light","Heavy","Heavy","Heavy","Light","Medium","Heavy","Heavy","Light","Heavy","Heavy","Heavy","Heavy","Heavy","Heavy","Light","Medium","Medium","Light","Heavy","Light","Light","Light","Light","Light","Heavy","Light","Medium","Heavy","Heavy","Heavy")
names(map_vector)<- type_of_weather
ingolstadt_weather = ingolstadt_weather %>%
mutate(season = ifelse(month(date) %in% c(12,1,2),"winter",NA)) %>%
mutate(season = ifelse(month(date) %in% c(3,4,5),"spring",season)) %>%
mutate(season = ifelse(month(date) %in% c(6,7,8),"summer",season)) %>%
mutate(season = ifelse(month(date) %in% c(9,10,11),"autumn",season))# %>% dplyr::select(-tsun)
day_description_impact = weatherstack_kelheim_daily %>% pivot_wider(names_from = description,values_from = n)
#remove NAs
day_description_impact[is.na(day_description_impact)] = 0
day_description_impact = day_description_impact %>% pivot_longer(cols = all_of(type_of_weather),names_to = "description",values_to = "value")
day_description_impact = day_description_impact
day_description_impact$description = map_vector[(day_description_impact$description)]
day_description_impact= day_description_impact %>% group_by(date)%>%
top_n(n = 1,value) %>% group_by(date) %>% top_n(n = 1,description) %>% rename(weather_impact = value)
#####Join the data#####
result_data = demand %>% inner_join(day_description_impact, by = "date") %>% inner_join(ingolstadt_weather,by = "date") %>% inner_join(df_stringency,by = "date") %>% mutate(date = as.Date(date,format = "%Y-%m-%d"))
#Also need to be added weekday
result_data = result_data %>% mutate(wday = as.character(wday(date,week_start = 1)))
#Append holidays
result_data = result_data %>% left_join(df_holidays, by = "date") %>% replace_na(list(isHoliday = FALSE,snow = 0)) #%>% filter(noRides != 0) #%>% filter(date <"2021-07-01")
head(result_data)
From following plots we can see strong correlation between day of the week and number of Rides, because we want to simulate common day using MatSim it was decided to filter weekends out of resulting dataset for the model. Also because day doesn’t represent weather impact, like holidays that have strong impact on mobility so they are also excluded. First 4 days are excluded, because they have 0 rides due to KeXi service start.
wday_plot = ggplotly(ggplot(result_data)+
geom_boxplot(aes(x = wday,y = noRides)))
holiday_plot = ggplotly(ggplot(result_data)+
geom_boxplot(aes(x = isHoliday,y = noRides )))
annotations = list(
list(
x = 0.2,
y = 1.0,
text = "Weekday",
xref = "paper",
yref = "paper",
xanchor = "center",
yanchor = "bottom",
showarrow = FALSE
),
list(
x = 0.75,
y = 1.0,
text = "Is Holiday",
xref = "paper",
yref = "paper",
xanchor = "center",
yanchor = "bottom",
showarrow = FALSE
))
subplot(wday_plot,holiday_plot) %>% layout(annotations = annotations)
result_data = result_data %>% filter(wday!=6 & wday!=7,isHoliday == FALSE, noRides!=0)
After first data processing it would be helpful to find some dependencies in the data using scatter plots mapped to number of rides. Here is summary of end dataset
result_data$description = factor(result_data$description)
result_data$season = factor(result_data$season)
summary(result_data)
## date noRides noRequests avgEuclidianDistance_m avgTravelTime_s description
## Min. :2020-06-30 Min. : 1.00 Min. : 4.0 Min. : 716.7 Min. : 0.0 Clear : 6
## 1st Qu.:2020-10-31 1st Qu.: 63.50 1st Qu.:132.0 1st Qu.:1080.7 1st Qu.:370.2 Cloudy:175
## Median :2021-03-15 Median : 82.00 Median :161.0 Median :1225.3 Median :426.6 Heavy : 18
## Mean :2021-04-06 Mean : 84.14 Mean :169.9 Mean :1565.2 Mean :351.2 Light :125
## 3rd Qu.:2021-09-14 3rd Qu.:107.00 3rd Qu.:198.5 3rd Qu.:2160.0 3rd Qu.:466.8 Medium: 6
## Max. :2022-01-28 Max. :150.00 Max. :408.0 Max. :4689.6 Max. :584.6 Sunny : 24
## weather_impact tavg tmin tmax prcp snow
## Min. : 4.0 Min. :-9.100 Min. :-13.900 Min. :-5.400 Min. : 0.000 Min. : 0.0000
## 1st Qu.: 9.0 1st Qu.: 2.700 1st Qu.: -0.350 1st Qu.: 5.625 1st Qu.: 0.000 1st Qu.: 0.0000
## Median :12.0 Median : 8.250 Median : 4.300 Median :13.150 Median : 0.000 Median : 0.0000
## Mean :12.9 Mean : 9.209 Mean : 5.029 Mean :13.565 Mean : 1.627 Mean : 0.5085
## 3rd Qu.:17.0 3rd Qu.:16.275 3rd Qu.: 11.600 3rd Qu.:21.175 3rd Qu.: 1.100 3rd Qu.: 0.0000
## Max. :24.0 Max. :25.100 Max. : 18.000 Max. :34.200 Max. :48.900 Max. :50.0000
## wdir wspd wpgt pres tsun season stringency
## Min. : 7.0 Min. : 1.400 Min. : 7.60 Min. : 985.4 Mode:logical autumn:123 Min. :37.04
## 1st Qu.:107.5 1st Qu.: 5.000 1st Qu.: 20.50 1st Qu.:1014.2 NA's:354 spring: 43 1st Qu.:55.09
## Median :230.0 Median : 6.650 Median : 27.70 Median :1018.9 summer: 91 Median :62.04
## Mean :195.1 Mean : 7.974 Mean : 30.72 Mean :1018.9 winter: 97 Mean :63.37
## 3rd Qu.:258.0 3rd Qu.: 9.625 3rd Qu.: 38.90 3rd Qu.:1024.4 3rd Qu.:75.00
## Max. :356.0 Max. :29.200 Max. :100.10 Max. :1043.4 Max. :85.19
## wday isHoliday
## Length:354 Mode :logical
## Class :character FALSE:354
## Mode :character
##
##
##
tavg_plot = ggplotly(
ggplot(result_data)+
geom_point(aes(y= noRides,x = tavg,colour = season))
)
pres_plot= ggplotly(
ggplot(result_data)+
geom_point(aes(y= noRides,x = pres,colour = season))
)
prcp_plot = ggplotly(
ggplot(result_data %>% filter(prcp!=0))+
geom_point(aes(y= noRides,x = prcp,colour = season))
)
snow_plot = ggplotly(
ggplot(result_data %>% filter(snow!=0))+
geom_point(aes(y= noRides,x = snow,colour = season))
)
subplot(tavg_plot,pres_plot,prcp_plot,snow_plot,nrows = 2)
After looking at these plots we can barely see existing strong dependencies or clusters, but there is one conspicious fact, that only during summer number of rides falls under ~50 while other seasons have more than 50 rides.
Correlation coefficients between two variables we calculate using cor() function, later anova oneway test will be used for testing to check categorical variable dependency.
best_pred <- result_data %>% ungroup() %>%
dplyr::select(-noRides,-description ,-date,-season,-noRequests,-avgEuclidianDistance_m,-avgTravelTime_s,-wday) %>%
map_dbl(cor,y = result_data$noRides) %>%
#map_dbl(abs) %>%
sort(decreasing = TRUE)
print(best_pred)
## pres snow wspd wdir prcp wpgt stringency
## 0.18474574 0.10739362 0.03762682 0.00967347 -0.01493169 -0.06206359 -0.12333568
## weather_impact tmin tavg tmax
## -0.21807896 -0.31030789 -0.36408674 -0.37219442
As a result highest absolute score in correlation have temperature(max), snow, pressure(air pressure), stringency(strictness of covid19 policy), weather impact(artificial made variable of highest weather description hours).
Testing of null hypothesis is made for dependent noRides and independent season, description and wday
season_test = oneway.test(result_data$noRides~result_data$season)$p.value
description_test = oneway.test(result_data$noRides~result_data$description)$p.value
wday_test = oneway.test(result_data$noRides~result_data$wday)$p.value
print(c("season" =season_test,"description" =description_test,"wday" =wday_test))
## season description wday
## 3.323634e-08 3.595709e-05 8.941111e-01
From the p-value we can see that null hypothesis can only be rejected for wday (p.value higher than 0.05), for the description and season we can’t make such conclusion. But one thing that should be also tested is dependency between seasons and temperature - 3.2481382^{-75}. As we can see temperature dependent from season, so in our future model one of the variable should be dropped.
After determination of significant variables we can build different models using different set of variables, one thing remains same - first approach is to build linear model to make predictions, also to check which predictors impact number of rides most. In our approach we will firstly larger model using all relevant variables from previous chapter, then we reduce number of variables taking smaller subset and synchronous to check how model quality changes.
data = result_data
omega_model = lm(noRides ~ tavg+pres+stringency+snow+weather_impact*description,data = data)
summary(omega_model)
##
## Call:
## lm(formula = noRides ~ tavg + pres + stringency + snow + weather_impact *
## description, data = data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -59.680 -17.402 -2.459 17.193 69.934
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 30.0833 196.6141 0.153 0.8785
## tavg -1.4445 0.2392 -6.040 4.06e-09 ***
## pres 0.2128 0.1880 1.132 0.2584
## stringency -0.6841 0.1376 -4.972 1.06e-06 ***
## snow 0.3732 0.4395 0.849 0.3964
## weather_impact -9.4743 6.5340 -1.450 0.1480
## descriptionCloudy -96.6944 64.2373 -1.505 0.1332
## descriptionHeavy -105.1883 68.2478 -1.541 0.1242
## descriptionLight -109.3593 64.2148 -1.703 0.0895 .
## descriptionMedium -113.1355 72.3294 -1.564 0.1187
## descriptionSunny -92.1232 74.9283 -1.229 0.2197
## weather_impact:descriptionCloudy 8.6618 6.5467 1.323 0.1867
## weather_impact:descriptionHeavy 10.1402 6.9605 1.457 0.1461
## weather_impact:descriptionLight 9.8958 6.5642 1.508 0.1326
## weather_impact:descriptionMedium 10.5065 7.6842 1.367 0.1724
## weather_impact:descriptionSunny 9.3651 7.4120 1.264 0.2073
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 26.78 on 338 degrees of freedom
## Multiple R-squared: 0.2574, Adjusted R-squared: 0.2245
## F-statistic: 7.811 on 15 and 338 DF, p-value: 4.418e-15
Low adjusted R^2 doesn’t tells that our model doesn’t fit the data. Let’s take a look on predicted values and residuals.
colors = c("actual" = "blue","predicted" = "red","residuals" = "gray50","zerorides" = "purple")
model = omega_model
test_data = data %>% add_predictions(model = model) %>% add_residuals(model = model) %>% mutate(error = ifelse(abs(resid)>=20,"extreme","normal"))
ggplotly(ggplot(test_data %>% filter(year(date)>=2020)) +
geom_line(aes(x = date,y = noRides,color = "actual"))+
geom_line(aes(x = date,y = pred,color = "predicted"))+
geom_line(aes(x = date,y = resid,color = "residuals"))+
geom_ref_line(h = 0)+
scale_color_manual(values = colors))
As we can see from our residuals plot our data has continuous growing trend, that our model doesn’t consider, so taking a date variable to model can drastically improve model “quality”.
omega_date_model = lm(noRides ~ tavg+pres+stringency+snow+weather_impact*description+date+season,data = data)
summary(omega_date_model)
##
## Call:
## lm(formula = noRides ~ tavg + pres + stringency + snow + weather_impact *
## description + date + season, data = data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -81.141 -8.559 -0.059 10.232 39.732
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -2.340e+03 1.532e+02 -15.271 < 2e-16 ***
## tavg -2.652e-01 2.220e-01 -1.195 0.233
## pres -1.044e-01 1.141e-01 -0.915 0.361
## stringency -6.280e-02 1.202e-01 -0.522 0.602
## snow 1.964e-01 2.635e-01 0.746 0.456
## weather_impact -2.291e+00 3.924e+00 -0.584 0.560
## descriptionCloudy -1.913e+01 3.859e+01 -0.496 0.621
## descriptionHeavy -2.759e+01 4.097e+01 -0.674 0.501
## descriptionLight -2.139e+01 3.858e+01 -0.554 0.580
## descriptionMedium -2.452e+01 4.342e+01 -0.565 0.573
## descriptionSunny -3.205e+01 4.500e+01 -0.712 0.477
## date 1.372e-01 5.631e-03 24.364 < 2e-16 ***
## seasonspring -6.217e+00 4.052e+00 -1.534 0.126
## seasonsummer -1.223e+01 3.058e+00 -4.001 7.77e-05 ***
## seasonwinter -5.933e+00 3.622e+00 -1.638 0.102
## weather_impact:descriptionCloudy 1.852e+00 3.933e+00 0.471 0.638
## weather_impact:descriptionHeavy 3.100e+00 4.175e+00 0.742 0.458
## weather_impact:descriptionLight 1.615e+00 3.944e+00 0.409 0.682
## weather_impact:descriptionMedium 1.830e+00 4.617e+00 0.396 0.692
## weather_impact:descriptionSunny 2.684e+00 4.456e+00 0.602 0.547
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 15.99 on 334 degrees of freedom
## Multiple R-squared: 0.7384, Adjusted R-squared: 0.7235
## F-statistic: 49.61 on 19 and 334 DF, p-value: < 2.2e-16
colors = c("actual" = "blue","predicted" = "red","residuals" = "gray50","zerorides" = "purple")
model = omega_date_model
test_data = data %>% add_predictions(model = model) %>% add_residuals(model = model) %>% mutate(error = ifelse(abs(resid)>=20,"extreme","normal"))
ggplotly(ggplot(test_data %>% filter(year(date)>=2020)) +
geom_line(aes(x = date,y = noRides,color = "actual"))+
geom_line(aes(x = date,y = pred,color = "predicted"))+
geom_line(aes(x = date,y = resid,color = "residuals"))+
geom_ref_line(h = 0)+
scale_color_manual(values = colors))
As expected R^2 squared increased as well residual standard error is decreased to a number of 15,99. ~0.7 of R-squared is relatively high in weather statistics. To check correctness of used approach residuals have to be normally distributed.
Residuals histogram
barplot <- ggplot(test_data, aes(x = resid ))+
geom_histogram(aes(y = stat(density)),colour="black", fill="white", binwidth=7)+
geom_density( fill="#FF6666",adjust = 10,alpha = 0.5) +
ggtitle("Omega model residuals")
ggplotly(barplot)
normal_dist = fitdistrplus::fitdist(test_data$resid,"norm")
plot(normal_dist)
From described plots we can see that our residuals looks normally
distributed with skewness to left tail, with some outliers. But simple
Shapiro-Wilk test gives p-value of 1.2434399^{-8} out, so our residuals
aren’t
Let’s take some variables out and see how model performs on the data and F Statistics. Low correlating variables from “Finding correlations” chapter are snow, stringency and weather_impact multiplied by description. Also there is strong correlation between average temperature and a season so our new model should contain only 1 of the predictors, because temperature have most impact on mobility and more large-scaled (season contains only 4 factors) we will take season out from new reduced_model.
reduced_1_model = lm(noRides ~ tavg+pres+stringency+snow+weather_impact*description+date,data = data)
summary(reduced_1_model)
##
## Call:
## lm(formula = noRides ~ tavg + pres + stringency + snow + weather_impact *
## description + date, data = data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -81.654 -9.119 -0.042 10.455 38.249
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -2.297e+03 1.544e+02 -14.885 < 2e-16 ***
## tavg -6.987e-01 1.491e-01 -4.687 4.04e-06 ***
## pres -8.945e-02 1.153e-01 -0.776 0.438305
## stringency -2.842e-01 8.552e-02 -3.323 0.000987 ***
## snow 1.428e-01 2.681e-01 0.533 0.594640
## weather_impact -3.033e+00 3.992e+00 -0.760 0.447908
## descriptionCloudy -2.463e+01 3.927e+01 -0.627 0.530916
## descriptionHeavy -3.495e+01 4.171e+01 -0.838 0.402630
## descriptionLight -2.651e+01 3.930e+01 -0.675 0.500348
## descriptionMedium -3.054e+01 4.422e+01 -0.691 0.490287
## descriptionSunny -4.654e+01 4.571e+01 -1.018 0.309324
## date 1.351e-01 5.644e-03 23.930 < 2e-16 ***
## weather_impact:descriptionCloudy 2.650e+00 3.999e+00 0.663 0.507911
## weather_impact:descriptionHeavy 3.955e+00 4.251e+00 0.930 0.352836
## weather_impact:descriptionLight 2.305e+00 4.014e+00 0.574 0.566184
## weather_impact:descriptionMedium 2.617e+00 4.696e+00 0.557 0.577605
## weather_impact:descriptionSunny 4.258e+00 4.523e+00 0.941 0.347214
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 16.32 on 337 degrees of freedom
## Multiple R-squared: 0.7249, Adjusted R-squared: 0.7118
## F-statistic: 55.5 on 16 and 337 DF, p-value: < 2.2e-16
colors = c("actual" = "blue","predicted" = "red","residuals" = "gray50","zerorides" = "purple")
model = reduced_1_model
test_data = data %>% add_predictions(model = model) %>% add_residuals(model = model) %>% mutate(error = ifelse(abs(resid)>=20,"extreme","normal"))
ggplotly(ggplot(test_data %>% filter(year(date)>=2020)) +
geom_line(aes(x = date,y = noRides,color = "actual"))+
geom_line(aes(x = date,y = pred,color = "predicted"))+
geom_line(aes(x = date,y = resid,color = "residuals"))+
geom_ref_line(h = 0)+
scale_color_manual(values = colors))
As we can see excluding variables from model doesn’t make it worse, we can also use anova test function from stats package to check model varable significance. But firstly we would like to get best best model with possible minimum variables. For this purpose we can inspect p-value from model summary, it says how significant separate variable in model predictions, less p-value - more significant. So we would like to exclude snow, and combined term ow weather_impact and description out of model.
reduced_2_model = lm(noRides ~ tavg+pres+stringency+weather_impact+description+date,data = data)
summary(reduced_2_model)
##
## Call:
## lm(formula = noRides ~ tavg + pres + stringency + weather_impact +
## description + date, data = data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -81.520 -9.297 0.208 10.323 38.075
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -2.284e+03 1.473e+02 -15.502 < 2e-16 ***
## tavg -7.061e-01 1.460e-01 -4.837 2e-06 ***
## pres -1.229e-01 1.104e-01 -1.113 0.266553
## stringency -2.897e-01 8.475e-02 -3.418 0.000707 ***
## weather_impact -4.058e-01 1.974e-01 -2.056 0.040580 *
## descriptionCloudy 8.102e-01 7.102e+00 0.114 0.909234
## descriptionHeavy 2.954e+00 7.914e+00 0.373 0.709175
## descriptionLight -4.415e+00 7.026e+00 -0.628 0.530141
## descriptionMedium -3.800e+00 9.626e+00 -0.395 0.693255
## descriptionSunny -3.243e+00 7.435e+00 -0.436 0.662972
## date 1.349e-01 5.541e-03 24.335 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 16.24 on 343 degrees of freedom
## Multiple R-squared: 0.7227, Adjusted R-squared: 0.7146
## F-statistic: 89.39 on 10 and 343 DF, p-value: < 2.2e-16
colors = c("actual" = "blue","predicted" = "red","residuals" = "gray50","zerorides" = "purple")
model = reduced_2_model
test_data = data %>% add_predictions(model = model) %>% add_residuals(model = model) %>% mutate(error = ifelse(abs(resid)>=20,"extreme","normal"))
ggplotly(ggplot(test_data %>% filter(year(date)>=2020)) +
geom_line(aes(x = date,y = noRides,color = "actual"))+
geom_line(aes(x = date,y = pred,color = "predicted"))+
geom_line(aes(x = date,y = resid,color = "residuals"))+
geom_ref_line(h = 0)+
scale_color_manual(values = colors))
As we can see removing snow and separation of weather_impact and description, increased model quality with reducing residual standard error. Anova test between these two models also doesn’t show significant difference 0.8449078 because p-value is 0.85>>0.05.
After few iterations we get to a model that contains only strictness of the policy restrictions, temperature and data. Represented model explains also explains most of the observations with relatively low residual standard error, but it is minimal for excluding, because anova significance difference between models without any of remaining predictors has p-value<0.05. So we get to our final model, that explains number of KeXi Rides using temperature, covid strictness level and date(positive growing trend).
reduced_3_model = lm(noRides ~ season+tavg+stringency+date,data = data)
summary(reduced_3_model)
##
## Call:
## lm(formula = noRides ~ season + tavg + stringency + date, data = data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -83.191 -8.784 -0.064 9.938 41.926
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -2.452e+03 1.035e+02 -23.695 < 2e-16 ***
## seasonspring -6.608e+00 3.898e+00 -1.695 0.090938 .
## seasonsummer -1.172e+01 3.007e+00 -3.899 0.000116 ***
## seasonwinter -6.147e+00 3.494e+00 -1.760 0.079371 .
## tavg -3.498e-01 2.031e-01 -1.722 0.085948 .
## stringency -3.222e-02 1.171e-01 -0.275 0.783393
## date 1.360e-01 5.386e-03 25.251 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 15.99 on 347 degrees of freedom
## Multiple R-squared: 0.7281, Adjusted R-squared: 0.7234
## F-statistic: 154.9 on 6 and 347 DF, p-value: < 2.2e-16
colors = c("actual" = "blue","predicted" = "red","residuals" = "gray50","zerorides" = "purple")
model = reduced_3_model
test_data = data %>% add_predictions(model = model) %>% add_residuals(model = model) %>% mutate(error = ifelse(abs(resid)>=20,"extreme","normal"))
ggplotly(ggplot(test_data %>% filter(year(date)>=2020)) +
geom_line(aes(x = date,y = noRides,color = "actual"))+
geom_line(aes(x = date,y = pred,color = "predicted"))+
geom_line(aes(x = date,y = resid,color = "residuals"))+
geom_ref_line(h = 0)+
scale_color_manual(values = colors))
After performing a minimal model we also want to check once again how residuals are distributed and some additional statistics metrics.
barplot <- ggplot(test_data, aes(x = resid ))+
geom_histogram(aes(y = stat(density)),colour="black", fill="white", binwidth=9)+
geom_density( fill="#FF6666",adjust = 10,alpha = 0.5) +
ggtitle("Final model residuals")
ggplotly(barplot)
We see normal distributed variable with skewness to the legt with 3 outliers at the left (analysis of individual days shows, that extremely low number of rides were when operator collecting data has changed so this is data specific problem that could be ignored or excluded from model)
test_data = test_data %>% filter(resid>=-50)
normal_dist = fitdistrplus::fitdist(test_data$resid,"norm")
plot(normal_dist)
fitdistrplus::descdist(test_data$resid)
## summary statistics
## ------
## min: -41.1969 max: 41.92641
## median: 0.3115131
## mean: 0.6410021
## estimated sd: 14.30196
## estimated skewness: 0.1283846
## estimated kurtosis: 3.045112
So our residuals are normally distributed :)
Excluding more variables from a model doesn’t impove model as well increases residual error enormous, so we can make conclusion, that represented model is minimal fitting possible with variables that have most impact to number of rides by the weather. Remains only to check residuals of fitted model.
Performed LINEAR regression analysis shows, that there no strong correlations between number of rides and weather characteristics except of temperature, and temperature is also quite dependent from season, one difference between using season instead of the temperature is that relative temp changes in same season isn’t inspected, so it can be that relative fall of the temperature in summer also strongly impacts mobility. Also data was collected during covid-19 pandemie, so stringency-strictness is also significant by building an model. In the end date is used in the model, because there are positive growing trend with using KeXi ride service.